Homework 4 Solutions
Josh Hernandez
October 27, 2009
2.2  The Matrix Representation of a Linear Transformation
2.
Let
β
and
γ
be the standard ordered bases for
R
n
and
R
m
, respectively. For each linear transformation
T
:
R
n
→
R
m
, compute [
T
]
γ
β
b.
T
:
R
2
→
R
3
deﬁned by
T
(
a
1
,a
2
) = (2
a
1

a
2
,
3
a
1
+ 4
a
2
,a
1
).
Solution:
A transformation from a 2 to a 3 dimensional space has a 3
×
2 matrix:
[
T
]
γ
β
=
2 1
3
4
1
0
.
c.
T
:
R
3
→
R
deﬁned by
T
(
a
1
,a
2
,a
3
) = 2
a
1
+
a
2

3
a
3
.
Solution:
[
T
]
γ
β
=
(
2 1 3
)
.
4.
Deﬁne
T
:
M
×
2
×
2
(
R
)
→
P
2
(
R
) by
T
(
a b
c d
)
= (
a
+
b
) + (2
d
)
x
+
bx
2
.
Let
β
=
±
v
1
=
²
1 0
0 0
³
,v
2
=
²
0 1
0 0
³
,v
3
=
²
0 0
1 0
³
,v
4
=
²
0 0
0 1
³´
,
and
γ
=
{
1
,x,x
2
}
.
Solution:
T
(
v
1
) = 1,
T
(
v
2
) = 1 +
x
2
,
T
(
v
3
) = 0,
T
(
v
4
) = 2
x
. Thus
[
T
]
γ
β
=
1 1 0 0
0 0 0 2
0 1 0 0
.
5.
Let
α
±
v
1
=
²
1 0
0 0
³
,v
2
=
²
0 1
0 0
³
,v
3
=
²
0 0
1 0
³
,v
4
=
²
0 0
0 1
³´
,
and
β
=
{
1
,x,x
2
}
,
and
γ
=
{
1
}
.
c.
Deﬁne
T
:
M
×
2
×
2
(
F
)
→
F
by
T
(
A
) = trace(
A
). Compute [
T
]
γ
α
Solution:
T
(
v
1
) = 1,
T
(
v
2
) = 0,
T
(
v
3
) = 0,
T
(
v
4
) = 1. Thus
[
T
]
γ
α
=
(
1 0 0 1
)
.
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